Method of Operating a Furnace

ABSTRACT

A method of operating a furnace having process tubes and multiple burners where it is desired to conform the temperatures of the process tubes to selected target temperature criterion. The present method provides a systematic and quantitative approach to determine how to adjust burner flow rates to result in desired tube wall temperatures, for example to minimize the temperature deviation between tube wall temperatures at a predetermined elevation in the furnace.

BACKGROUND

The present disclosure is directed to operation of a furnace having aplurality of burners and containing a plurality of process tubes. Inparticular, the present disclosure is directed to operation of a furnacehaving a plurality of burners and containing a plurality of processtubes with improved efficiency.

Steam hydrocarbon (steam methane) reformers are furnaces containingnumerous process tubes (including configurations with more than 400reactor tubes), each tube containing catalyst (for example, a reformingcatalyst) for transporting a process fluid (for example, steam and ahydrocarbon). The furnace, for example, can include vertically extendingprocess tubes positioned in parallel rows with about 30 to 60 tubes ineach row. The distance between two rows of tubes is about 2 to 3 meters.The tubes can extend vertically about 12 meters and have an outerdiameter of 100 to 150 millimeters. The tubes can be positioned in therow with a center-to-center spacing of 250 to 500 mm. About 10 to 20burners can be positioned between each set of two rows of tubes. A totalof eight or more rows of tubes and nine or more rows of burners can beincluded in a furnace.

Generally, energy efficiency for industrial processes is becoming moreimportant. For many processes, such as hydrogen production, theefficiency of the process is related to the ability to monitor andregulate certain temperatures. Measuring temperatures of reformer tubesin a reformer furnace can present several challenges. For example, whenmeasuring temperatures at specific locations on the reformer tubes,inconsistency in taking the measurements at the specific location of thereformer tube can result in measurements with greater uncertainty. Moreprecise monitoring of the temperature at the specific location on thereformer tube can permit improved energy efficiency by permitting moreaccurate data to be used for process control.

Furnace tube temperatures may vary along the length. The tubes may gethotter in the direction of the process flow as the process stream picksup heat from the furnace. Process tubes may cool due to endothermicreaction even as heat is added from the furnace. This axial variation ispart of the process. Traditional methods of furnace control require ameasure of temperature. This can be a tube wall temperature, a processgas temperature or a combustion gas temperature (or some combination).In traditional methods of furnace control, the overall flow of fuel (orin some cases oxidant or inerts) is adjusted to control the temperatureas described in U.S. Pat. Publ. US2007/0104641. Adjustments also may bemade to control the axial temperature profile.

Tube temperature may also vary from one tube to another. If there isaxial variation it is necessary to compare tubes at the same axialposition to determine the tube-to-tube variability. There may beoperational benefits to reducing the tube-to-tube variability or tocontrolling the variability. The methods described here are intended toaddress the issue of tube-to-tube variability or furnace balance. Thisis done in addition to the traditional control methods which adjust theoverall flow of fuel (or other stream) to control temperature.

Regulating temperatures in a furnace having process tubes and aplurality of burners for heating the process tubes can also presentseveral challenges. The complex interaction of flame heating from theplurality of burners coupled with the uncertainty of temperaturemeasurements has heretofore limited efficiency gains.

One way to improve the efficiency of a reformer furnace is to maintain auniformity of temperature among the process tubes at a predeterminedelevation in the furnace. Thus, the measuring or monitoring of thetemperature of each of the process tubes at a predetermined location orelevation can be needed to obtain the desired efficiency improvement. Inaddition, the process tubes of a furnace can be under very high internalpressures (up to about 50 atmospheres) and at very high temperatures (upto about 950° C.). Thus, a slight change in temperature can reduce theoperational life of a reactor tube. For example, operating at about 10°C. above the design temperature for the tube can reduce the operationallife of the tube by as much as one half. The cost of repairing and/orreplacing the tubes can be high due to the use of special alloys in thetubes that are needed to permit the tubes to survive the operationalconditions of the furnace. As such, furnace operators alsomeasure/monitor the tube temperatures to avoid exceeding the tube designtemperature in addition to trying to obtain efficiency improvements.

Industry desires to operate furnaces containing process tubes with auniformity of temperature among the process tubes at a predeterminedelevation in the furnace.

BRIEF SUMMARY

The present invention relates to a method of operating a furnace havinga plurality of burners, each of the plurality of burners having flowrates associated therewith, the furnace containing a plurality ofprocess tubes. The method comprises:

-   -   (a) selecting target temperature criterion for the plurality of        process tubes;    -   (b) measuring first temperature information comprising data for        each of the plurality of process tubes at a first operating        condition;    -   (c) providing an estimate of a mathematical function        characterizing a relationship between burner flow rate changes        for two or more of the plurality of burners and individual        temperature changes for at least a portion of the plurality of        process tubes;    -   (d) calculating a first target flow rate solution set having        solutions for the two or more of the plurality of burners        consistent with conforming temperatures of the plurality of        process tubes to the target temperature criterion for the        plurality of process tubes using the estimate of the        mathematical function and the first temperature information for        the plurality of process tubes; and    -   (e) adjusting a first valve upstream of at least one of the two        or more of the plurality of burners to change at least one of        the flow rates of the two or more of the plurality of burners in        accordance with the first target flow rate solution set wherein        the first valve is not upstream of all burners of the furnace.

The estimate of the mathematical function may be represented as

ΔT=GΔu

where ΔT represents the individual temperature changes for the at leasta portion of the plurality of process tubes, Δu represents burner flowrate changes for the two or more of the plurality of burners and G is again matrix.

In the method, the first temperature information for the plurality ofprocess tubes may be measured by:

-   -   capturing a first plurality of images of an interior area of the        reformer furnace, at least some images of the first plurality of        images being associated with different portions of the interior        area of the reformer furnace, wherein each image of the first        plurality of images comprises first pixel data associated with a        respective portion of the plurality of process tubes; and    -   processing a portion of the first pixel data to obtain the first        temperature information for the plurality of process tubes.

The method may further comprise:

-   -   measuring second temperature information for the plurality of        process tubes at a second operating condition different from the        first operating condition; and    -   wherein the estimate of the mathematical function provided in        step (c) is calculated using the first temperature information        and the second temperature information.

The second temperature information for the plurality of process tubesmay be measured by:

-   -   capturing a second plurality of images of the interior area of        the reformer furnace, at least some images of the second        plurality of images being associated with different portions of        the interior area of the reformer furnace, wherein each image of        the second plurality of images comprises second pixel data        associated with a respective portion of the plurality of process        tubes; and    -   processing a portion of the second pixel data to obtain the        second temperature information for the plurality of process        tubes.

The first temperature information may include uncertainty values and thesecond temperature information may include uncertainty values; and theestimate of the mathematical function and/or the updated estimate of themathematical function may be calculated using the uncertainty values ofthe first temperature information and the uncertainty values of thesecond temperature information.

The method may further comprise:

-   -   measuring second temperature information for the plurality of        process tubes at a second operating condition different from the        first operating condition and subsequent to the first operating        condition;    -   calculating a second target flow rate solution set having        solutions for the two or more of the plurality of burners        consistent with conforming temperatures of the plurality of        process tubes to the target temperature criterion using the        estimate or an updated estimate of the mathematical function and        using the second temperature information for the plurality of        process tubes; and    -   adjusting the first valve or a second valve upstream of at least        one of the two or more of the plurality of burners to change at        least one of the flow rates of the two or more of the plurality        of burners in accordance with the second target flow rate        solution set wherein the second valve is not upstream of all        burners of the reformer furnace.

The second operating condition may result from conducting step (e).

The method may further comprise:

-   -   updating the estimate of the mathematical function from the        second temperature information for the plurality of process        tubes thereby forming the updated estimate of the mathematical        function; and    -   the step of calculating the second target flow rate solution set        may then use the updated estimate of the mathematical function.

The estimate of the mathematical function may comprise calculated valuesfrom functional elements wherein each of the functional elementscomprise a functional form comprising a first functional parameter, asecond functional parameter and a geometric parameter characterizingdistances between each of the plurality of process tubes and each of theplurality of burners;

-   -   wherein the first functional parameter of a first functional        element of the functional elements has the same value as the        first functional parameter of a second functional element of the        functional elements; and    -   wherein the second functional parameter of a first functional        element of the functional elements has the same value as the        second functional parameter of a second functional element of        the functional elements.

The first functional parameter may have a value that is the same foreach of the functional elements and the second functional parameter mayhave a value that is the same for each of the functional elements.

The plurality of burners may comprise two or more rows of burners andthe first valve may be upstream of a first row of burners. The methodmay then further comprise:

-   -   measuring second temperature information for the plurality of        process tubes at a second operating condition different from the        first operating condition and subsequent to the first operating        condition;    -   calculating a second target flow rate solution set having        solutions for the two or more of the plurality of burners        consistent with conforming temperatures of the plurality of        process tubes to the target temperature criterion using the        estimate or an updated estimate of the mathematical function and        using the second temperature information for the plurality of        process tubes; and    -   adjusting a second valve upstream of a single burner of the two        or more of the plurality of burners to change at least one of        the flow rates of the single burner in accordance with the        second target flow rate solution set wherein the second valve is        not upstream of any burner other than the single burner.

The first valve may be upstream of a lone first burner of the two ormore of the plurality of burners. The method may then further comprise:

-   -   measuring second temperature information for the plurality of        process tubes at a second operating condition wherein the second        operating condition results from step (e);    -   adjusting a second valve in accordance with the first target        flow rate solution set wherein the second valve is upstream of a        lone second burner of the two or more of the plurality of        burners;    -   measuring third temperature information for the plurality of        process tubes at a third operating condition wherein the third        operating condition results from adjusting the second valve in        accordance with the first target flow rate solution set;    -   updating the estimate of the mathematical function from the        second temperature information and the third temperature        information thereby forming the updated estimate of the        mathematical function;    -   calculating a second target flow rate solution set having        solutions for the two or more of the plurality of burners        consistent with conforming temperatures of the plurality of        process tubes to the target temperature criterion using the        updated estimate of the mathematical function; and    -   adjusting at least one of the first valve, the second valve or a        third valve upstream of the two or more of the plurality of        burners to change at least one of the flow rates of the of the        two or more of the plurality of burner in accordance with the        second target flow rate solution set wherein the third valve is        not upstream of all burners in the reformer furnace.

BRIEF DESCRIPTION OF SEVERAL VIEWS OF THE DRAWINGS

FIG. 1 shows a cross-sectional view of a furnace.

FIG. 2 shows a representation of the fields of view of a camera used toacquire image data from the furnace.

FIG. 3 shows an exemplary contour plot of temperature differences for apredetermined elevation of a furnace.

FIG. 4 is a plot of an example function.

Wherever possible, the same reference numbers will be used throughoutthe drawings to represent the same parts.

DETAILED DESCRIPTION

The present invention will be described more fully hereinafter withreference to the accompanying drawings, in which an exemplary embodimentof the disclosure is shown. This disclosure may, however, be embodied inmany different forms and should not be construed as limited to theembodiments set forth herein.

The articles “a” and “an” as used herein mean one or more when appliedto any feature in embodiments of the present invention described in thespecification and claims. The use of “a” and “an” does not limit themeaning to a single feature unless such a limit is specifically stated.The article “the” preceding singular or plural nouns or noun phrasesdenotes a particular specified feature or particular specified featuresand may have a singular or plural connotation depending upon the contextin which it is used. The adjective “any” means one, some, or allindiscriminately of whatever quantity.

As used herein, “plurality” means “at least two.”

The present invention relates to a method of operating a furnace wherethe furnace has a plurality of burners and contains a plurality ofprocess tubes. The furnace may have at least 10 burners, typicallybetween 20 and 500 burners. The burners are typically arranged in rows.The furnace may contain at least 20 process tubes, typically between 40and 1000 process tubes. The process tubes are typically arranged inrows. Temperatures in the furnace, notably reactor tube temperatures areregulated by adjusting fuel and/or oxidant flow rates through theburners.

The present method provides a systematic and quantitative approach todetermine how to adjust burner flow rates to result in desired tube walltemperatures, for example to minimize the temperature deviation betweentube wall temperatures at a predetermined elevation in the furnace.

FIG. 1 shows a cross-sectional representation of a furnace 102 with aplurality of process tubes 104 positioned within an interior area 106 offurnace 102. Furnace 102 may be a steam methane reformer, methanolreformer, ethylene cracker, platformer reheat furnace, radiant heatchamber, or other similar type of furnace, reformer or chamber. Theprocess tubes may contain a catalyst, for example, reforming catalyst.The catalyst may be in any form known in the art, for example, pelletsand structured packing. Process tubes 104 can be positioned in aplurality of rows 112 separated by burners 108. Process tubes 104 canextend either vertically or horizontally in furnace 102. A plurality ofburners 108 may be arranged in rows and used to raise the temperature ofthe interior area 106 of the furnace 102 to a desired temperature toaccomplish the process or activity to be performed inside the furnace102. The rows of tubes 104 and rows of burners 108 may be substantiallyparallel. Along the perimeter of furnace 102 are view ports 110 thatpermit tubes 104, burners 108 and any other structure or feature insidefurnace 102 to be viewed and/or analyzed from a point exterior to thefurnace 102. Pairs of view ports 110 may be positioned on the perimeterof the furnace 102 at opposite ends of a row of burners 108.

Oxidant gas flow rates and fuel flow rates are associated with each ofthe plurality of burners. Any known fuel or combination of fuels may beused, for example, natural gas or a by-product stream containing methaneand carbon monoxide from a pressure swing adsorber used to separatehydrogen from a reformer effluent. Oxidant gases include air, industrialgrade oxygen, oxygen-enriched air, and oxygen-depleted air such as gasturbine exhaust.

This method comprises selecting target temperature criterion (sometimesreferred in the art as an optimization target) for the plurality ofprocess tubes. This means that it is necessary to specify(mathematically) what constitutes the most preferred temperatures forthe plurality of process tubes. This target applies to the variabilityfrom tube-to-tube (the furnace balance) not along the tube length oraround the tube diameter. The most preferred temperatures are anidealization that cannot be realized exactly. So the target temperaturecriterion is a mathematical statement of which of the practicaltemperature variations are most preferred. The plurality of processtubes considered in the method need not include all of the process tubesin the furnace.

It may be preferred that there is no variability among the tubetemperatures. This is equivalent to saying that the measured temperatureof each tube is equal to the average temperature of all the tubes.Traditional furnace control allows the average temperature to beadjusted up or down to match a fixed temperature target and thistraditional control may be overlaid on top of the furnace balancing, butis not the basis for the target temperature criterion.

The target temperature criterion is formulated based on a measure ofvariability. Many measures of variability are in common usage includingthe variance and standard deviation, and other measures of variabilitymay be used (e.g., the maximum measured value minus the average measuredvalue). Combinations may be constructed which include these differentmeasures (e.g., a weighting factor times the variance plus anotherweighting factor times the difference between the maximum measuredtemperature and the average).

The temperature of tube j may be labeled T_(j) and the averagetemperature given by

$\overset{\_}{T} = {\frac{\sum\limits_{\forall j}T_{j}}{N_{tube}}.}$

An optimization target may be defined by the 2-norm, also called theL²-norm, associated with the difference between all T_(j) and T which wewrite symbolically as ∥(T− T)∥₂ where the underbar is used to designatea vector quantity. Here the vector of tube temperatures and

${x}_{p} = \sqrt[p]{\sum\limits_{\forall j}\left( {T_{j} - \overset{\_}{T}} \right)^{p}}$

defines the general p-norm and for which p=2 gives the 2-norm. Anothertarget temperature criterion may be defined as w∥(T− T)∥₂+(1−w)∥(T−T)∥_(∞) where w is a weighting factor with value between 0 and 1. Withw=1, this optimization function reduces to the optimization functionshown first (the 2-norm) and with w=0 this optimization function reducesto the difference between the hottest tube and the average tube. Theoptimization will be carried out to minimize the chosen optimizationtarget.

The target temperature criterion may also be combined with a temperatureoffset which is specified for individual tubes (Θ_(j) and with Θ thevector designating the set of all offsets and Θ the average offset).Symbolically, this would be ∥(T− T+( Θ−Θ))∥₂. This would allow, forexample, targeting uniform temperatures within one region of the furnacethat are different from uniform temperatures in another region. In thisform, specifying an offset for even a single tube adjusts the targettemperature of all the tubes through the average offset term and theoffset applies to the balance within the furnace without reference to aspecific average temperature which is controlled by traditional methods.

It may be desirable to specify upper or lower bounds on tubetemperatures. The optimization target may be specified to minimize thevariance and a traditional controller set to raise or lower the averagetemperature to satisfy the bounding condition.

The method further comprises measuring first temperature informationcomprising data for each of the plurality of process tubes at a firstoperating condition. Operating conditions include oxidant flow rates,fuel flow rates, reactant gas feed rates, hydrogen production rate,furnace pressure, etc.

Furnace 102 may have view ports 110 at one or more levels or elevations.Having view ports 110 at more than one level permits greater viewcapability of the tubes 104.

The first temperature information of the plurality of process tubes maybe measured by capturing a first plurality of images of an interior areaof the reformer furnace, at least some images of the first plurality ofimages being associated with different portions of the interior area ofthe reformer furnace, wherein each image of the first plurality ofimages comprises first pixel data associated with a respective portionof the plurality of process tubes, and processing a portion of the firstpixel data to obtain the first temperature information for the pluralityof process tubes.

Temperature information may be obtained by taking a series of digitalimages of the tubes 104 in furnace 102 through view ports 110. Thedigital images may be taken by a digital camera or any other devicecapable of capturing digital image information. The digital images maybe “still” digital images from a video camera (i.e. a still image from acontinuous image device), or average digital images from a video camera(i.e. an image “averaged” over some time interval and not from just a“snapshot” at one time). The digital images may be obtained by pointingthe digital camera through each view port 110 and then capturing thecorresponding image information, i.e. “taking a picture” of thecorresponding portion of the interior area.

The digital camera may be positioned on a monopod or other similardevice to attempt to maintain desired pitch, yaw, and roll angles in thedigital images and to assist in positioning the lens of the camera inthe center of the view port 110. Further, the camera may be set to amanual mode for maintaining a more consistent aperture speed and thefocus may be set to infinity.

To obtain an image of the interior area 106 of the furnace 102, whichincludes the tubes 104, the camera may be briefly placed up to a viewport 110 with the operator pointing the camera through the view port 110and subsequently depressing the shutter button to capture the image andthen removing the camera from view port 110. FIG. 2 shows the fields ofview 120 of the camera when taking images from several view ports 110.As can be seen in FIG. 2, each field of view of the camera includes aportion of one or more rows of tubes 104. The period of time that thecamera is pointing through the view port 110 should be minimized toprotect the camera and operator from excessive radiant heat exposure;however, the camera should not be moving when the shutter button isbeing depressed to ensure that the image is not blurred. The procedurecan be repeated for every view port 110 on the perimeter of furnace 102.

The camera may capture an image (or multiple images) through the viewport of a portion of the interior area 106 of the furnace 102. The imagemay include a row of tubes 104 located along the left side of the imageand another row of tubes 104 located along the right side of the image.In addition, the image may include an opposing view port. The opposingview port may be used to capture an image from the opposite wall offurnace 102. The image may include other structures and/or features ofthe interior area 106 of the furnace 102 that are within the field ofview of the camera.

The portion of the interior area 106 captured in an image may overlap orinclude similar portions of the interior area 106 captured in otherimages. Stated differently, the portion of the interior area 106 shownin one image may include structures or features that are also shown inother images. For example, images taken from adjacent viewports 110 mayshow opposite sides of the same tube 104. Similarly, images taken fromopposite viewports 110 may show the same tube 104 at different angles.Furthermore, the images are not required to correspond or map tospecific or exclusive regions of the interior area 106 and may showsubstantially similar regions or portions of the interior area 106. Animage shows a different portion of the interior area 106, if the imageincludes one structure or feature that is not shown in another image orshows the same structures or features at different angles orperspectives than the other images.

Images of the interior area 106 and tubes 104 from each view port 110may be captured according to a predetermined sequence or along apredetermined path 202 around the perimeter of the furnace 102 as shownin FIG. 2. Predetermined path 202 can extend in either a clockwise orcounter-clockwise direction. By capturing images in a predeterminedsequence, the subsequent identification of the portion of interior area106 captured in each image can be quickly accomplished since each stepof the sequence or path corresponds to a known portion of interior area106. Images of interior area 106 and tubes 104 may be taken in anydesired order or sequence with the additional requirement that thecorresponding portion of interior area 106 be correlated with thecaptured image. Since furnace 102 can include view ports 110 on oppositesides of furnace 102 and on each side of a row of tubes 104, all of thetubes 104 can be included in at least two images and many of the tubes104 can be included in at least four images.

A single camera may be used to capture all of the images of the interiorarea 106 of the furnace 102. Utilizing the single camera to capture allof the images may increase the consistency of subsequent processing andanalysis of the image data because the images are captured under uniformcamera conditions such as uniform signal to noise levels and uniformsensitivities to different wavelengths. However, a plurality of camerasmay be used to capture images, but subsequent processing and analysis ofthe image data should account for differences in the camera conditionssuch as differences in the sensitivities to different wavelengths andthe differences in signal to noise ratios as a result of differences inconditions between cameras and/or models of cameras. The accounting fordifferences in camera conditions is needed to make image data acquiredfrom one camera correspond with image data acquired from another camera.

When capturing an image of the interior area 106 of the furnace 102, thecamera may include one or more filters to prevent or reduce certainwavelengths of light from reaching the imager or sensor. The image orsensor can include charge-coupled devices (CCDs) and/or complementarymetal-oxide semiconductor (CMOS) devices. The filter may be designed topermit 50% of the light at a predetermined wavelength to pass throughthe filter and reach the sensor. The filter may be further designed topermit less light, i.e., less than 50% of the light, to pass through thefilter at wavelengths that are less than the predetermined wavelength,and to permit more light, i.e., greater than 50% of the light, to passthrough the filter at wavelengths that are greater than thepredetermined wavelength. The predetermined wavelength may be about 715nm or the predetermined wavelength may be between about 300 nm or lessand/or 1000 nm or more.

The imager or sensor that is incorporated into the camera can includepixels that record the intensity of light received at the pixel. Thenumber of pixels in the camera corresponds to the resolution of thecamera. The camera may have a resolution between about 1 megapixel(approximately 1 million pixels) to about 10 megapixels (approximately10 million pixels) or more. Each pixel in the imager or sensor may haveone or more components or channels that record the intensity of light.Each pixel of the imager or sensor can have three components orchannels, which may correspond to red (R), green (G) and blue (B)colors. The channels or components of the pixel can be configured to bemore receptive to light at certain predetermined frequencies and lessreceptive to light at other predetermined frequencies when recording theintensity of light. In other words, light at certain predeterminedfrequencies can contribute more to the overall intensity measurementthan light at other predetermined frequencies. When an image iscaptured, the recorded intensities for each channel or component of eachpixel are stored as image data or pixel data. The pixels may beconfigured to measure the intensity of light in the visible spectrum.

After the images of the furnace 102 are obtained, the correspondingimage data for each of the images may be loaded into a computer or otherprocessing device for additional processing and analysis. Each of theimages may then be processed using the computer to correct, i.e., reduceor eliminate any distortion in the image. Before each image can beprocessed to correct for distortion in the image, a transformation modelto represent each lens and camera combination used to capture images maybe constructed or created. To create a transformation model, a series ofradial distortion models may be created for the lens and cameracombination. A radial distortion model determines the amount of radialdistortion that may be introduced by a calibrated lens and cameracombination for a selected focal length (accounting for the possibilityof a zoom lens) and selected focal distance. Once the transformationmodel has been created for a lens and camera combination, thetransformation model can be applied to the images captured by that lensand camera combination to correct for any distortion. Methods fordistortion correction are known in the art. Any suitable method fordistortion correction may be used in connection with obtainingtemperature information.

In addition to distortion correction, each of the images may beprocessed using the computer to correct for any rotation or deviation(“rotation correction”) from a specific position, for example, a“centered position.” Rotation correction can be performed to correct thevertical orientation of the image (“roll angle”), to correct the “up anddown” angle (“pitch”) and the “left and right” angle (“yaw”). The rollangle and pitch may be corrected by detecting the edge(s) of the lasttube(s) 104 in the image and the position of the opposite view port 110and then adjusting the image so the edge(s) of the last tube(s) 104 inthe image are vertical. Yaw may be corrected using furnace geometryinformation. Methods for rotation correction are known in the art. Anysuitable method for rotation correction may be used in connection withobtaining temperature information.

The “corrected” images may be processed using the computer to detect ordetermine the edges of the tubes 104 and/or any other desirablefeatures, for example, opposing view port 110, in the image. Thedetected edges of the tubes 104 and the detected features from the imagemay be adjusted or modified in view of a geometric model of the furnace102. A geometric model is a representation of the position in space ofeach feature in the furnace (typically represented by x, y, zcoordinates and a reference point). Based on design specification suchas the row spacing, tube-to-tube spacing and the distance between thewall and the first tube, an “as-built” geometric model can be developed.Due to manufacturing tolerances and tube movement resulting from thermalexpansion, the tubes and other furnace features may not be located atprecisely the same position as in the “as-built” geometric model. Thismodel can be subsequently modified to match the actual operatingconditions of the furnace by comparing the edges detected in the imagesto the geometric model. This allows the tubes or other features to beidentified.

The geometrical model is used as the starting point to identify eachtube. Edge detection is used to fine-tune the location of the feature,because tubes can bow or move in a high-temperature environment.

The edge of the tube banks and location of the view port are estimatedbased on a modeling scheme that incorporates both the geometricalinformation and the result of the edge detection from the image. Theinformation of the edge detection from image processing is reconciledwith the geometrical data.

The edge detection algorithm or process estimates the possible locationof the edges and provides a probability distribution of where each edgemay be located. The probability distribution of the location of the edgeis compared with the geometrical model. Since there are multiple edgesthat are located at the same time, the error between the geometricalinformation and the probability density of the location of the edges areminimized to derive the final location of the edges.

Using the geometric model and the identified tube edges and otherfeatures, each tube 104 in each image may be identified and segmented.Starting with the detected edge of the last tube 104 in a row, a gridmay be overlaid onto a portion of the image corresponding to the tuberow. The grid may be unevenly spaced and may be based on dimensions andconfiguration from the geometric model such as the tube row spacing andtube center distance. The vertical lines of the overlaid grid correspondto the edges of tubes 104 based on known positions of tubes 104 from thegeometric model. The spacing between the grid lines can vary from 1pixel to 100 or more pixels depending on the resolution of the cameraused. The grid may include a row of segments having a predeterminedlength and/or height. The segments may be centered on a predeterminedelevation.

Each segment of the row of segments may then be checked to determine ifthe segment is part of a tube 104 or is another part of the interiorarea 106 as part of outlier or defect detection. If a segment is notpart of a tube 104, the segment is discarded. The remaining segments,which correspond to tubes 104 in furnace 102, are then used in theanalysis to determine a temperature of each of the tubes 104 in thecorresponding image. The method used to determine outliers or whether asegment is part of a tube is based on a classification tree. Theclassification tree is developed using information from the geometricmodel. A series of different statistics are tested for a segment andbased on the result of the tests, a segment is determined to be good(part of a tube) or bad (not a tube).

The temperature of the tubes 104 may be determined based on the pixeldata from all of the images. To determine a temperature value for a tube104 in an image, the pixel data in the segment of the corresponding tube104 is processed to determine a value representing a measure of centraltendency, which is then correlated to a temperature. The temperature ortemperature value is a representative value for a tube. The tubetemperature varies along its length and one or more selected elevationsare measured to provide representative temperature value(s) for a tube.The processing of the pixel data, for example, intensity values, beginswith obtaining a value representing a measure of central tendency, foreach channel or component, from the pixel data of the pixels in thesegment. The value representing a measure of central tendency may be amedian of the pixel data. However, in other embodiments, the valuerepresenting a measure of central tendency may be a mean, robust mean,mode or other statistical measure derived from the pixel data. The valuerepresenting a measure of central tendency for each channel or componentmay then be correlated to a temperature value. The temperature value fora segment determined from the value representing a measure of centraltendency may also be assigned an uncertainty value. The uncertaintyvalue can quantify the confidence in the determined temperature value.Numerous factors such as the distance of the tube from the camera (pathlength, “d”), the camera angle (formed by a center line of the cameraand the position of tube 104, “a”), the number of pixels in the segmentrepresenting the tube, can affect the confidence of a temperature valuedetermination. If the pixel data includes multiple channels orcomponents, the temperature value for each of the channels or componentscan be averaged using a statistical averaging technique, for example,average, weighted average, etc., to obtain a single temperature valuefor the segment, which corresponds to a tube 104.

To obtain a correlation between temperature values and the pixel data, arelationship between known temperatures and pixel data may be formed andstored in a database or other memory device for accessibility in thefuture. One technique to obtain the relationship between pixel data andtemperature involves capturing an image of the interior area 106 andthen soon thereafter following the image capture with the acquisition oftemperature measurements of the tubes in the portion of the interiorarea corresponding to the image. The temperature measurements of thetubes may be performed with an optical pyrometer or other suitabledevice. The values representing a measure of central tendency from theimage, which correspond to tubes 104, are then compared to thetemperature measurements to establish a correlation or relationshipbetween temperature and pixel value. Other techniques to obtaintemperature information on the tubes 104 can also be used to establishthe relationship or correlation to pixel values. Once a relationship orcorrelation between temperatures and pixel values is established, thecorrelation can be used for subsequent processing of images.

Once the temperature values for each tube 104 in each image isdetermined, the temperature values from the images can be combinedtogether to provide temperature information on all the tubes 104 infurnace 102. The temperature value from each tube 104 in each image isextracted and used to generate a representation of temperatureinformation for all of the tubes 104 of the furnace 102. Where aparticular tube 104 has several temperature values as a result of thetube 104 being in more than one image, the temperature values may beaveraged using a statistical averaging technique, for example, average,weighted average, etc. The uncertainty of the temperature values may beincluded as a factor when calculating a weighted average. Once theextraction and processing of the temperature values from the images iscomplete, a temperature value for each tube 104 in the furnace 102 maybe displayed.

Instead of determining a temperature value for each tube 104 in eachimage, the segment pixel data or the values representing a measure ofcentral tendency may be continued to be processed in a manner similar tothat discussed above (including the application of uncertainty values)to generate a representation of the furnace 102 in pixel data orstatistical values. The pixel data or values representing a measure ofcentral tendency in the representation of the furnace can then beconverted to temperatures using the relationship or correlationdiscussed above to obtain a representation of the furnace based ontemperature values.

A multivariate regression method (such as Partial Least Squares) may beused to establish a correlation between the temperature of the specifictubes for which independent temperature measurements are available andthe pixel data from an image. The correlation can incorporate variablesincluding, but not limited to, channel pixel values, for example, R, G,B values, d, a, other suitable quantifiable measurements, and/orcombinations thereof. For example, the correlated value of the tubetemperature can be represented as {circumflex over (T)}_(j) (for tube j)and the independent variables as x_(ij), where i denotes the i^(th)variable, from a partial list of variables where

$x \in {\left\{ {R,G,B,\frac{1}{d},\frac{1}{d^{2}},\alpha,\ldots} \right\}.}$

Other variables associated with the tube temperature may include the R,G, B of the previous and next tubes. The tube 104 temperature at apredetermined location can be represented as a linear combination ofthese variables with unknown coefficients A_(i) such that {circumflexover (T)}_(j)=Σ_(i)A_(i)x_(ij). Given a set of independent temperaturemeasurements, T_(j), where j=1, 2, . . . , n, which can come from apyrometer, the unknown coefficients can be determined by minimizing theerror between the actual data and the prediction in the least squaressense:

${\underset{{j = 1},n}{Min}\left\{ \left( {T_{j} - {\hat{T}}_{j}} \right)^{2} \right\}} = {\underset{{j = 1},n}{Min}{\left\{ \left( {T_{j} - {\sum\limits_{i}{A_{i}x_{ij}}}} \right)^{2} \right\}.}}$

These evaluations can be systematically performed with the aid ofstandard statistical and mathematical software tools (for example,Matlab®). The final result of the evaluations can generate a correlationbetween data from image and temperatures of tubes 104 in the leastsquares sense {circumflex over (T)}_(j)=Σ_(i)A_(i)x_(ij) allowingtemperature estimates for all tubes in the images (not just those forwhich independent temperature measurements are available).

Referring to FIG. 3, temperature information regarding the tubes 104 offurnace 102 may be displayed as a contour plot or other suitable (color)graphic representation. FIG. 3 shows an exemplary contour plot oftemperature difference information for the tubes 104 of a furnace 102 ata predetermined elevation. The plot can identify individual processtubes and rows. As shown, the plot illustrates that the furnace includesregions of above-average temperature 502, regions of below-averagetemperature 504, and regions of average temperature 506.

The process for obtaining temperature information disclosed herein maybe applied to a plurality of elevations within furnace 102 and may beused to generate a three-dimensional view or representation oftemperature data. View ports 110 can be located in upper and lowerportions of furnace 102. Performing the process discussed above withboth view ports in the upper and lower portions of furnace 102 permitsthe generation of a plot for the upper portion and the lower portion offurnace 102. Additional calculations incorporating anticipateddifferences in temperature at the various elevations may be incorporatedinto a three-dimensional plot. Incorporating the anticipated differencesin temperatures permits the plot to account for anomalies with specifictubes 104. Multiple rows of segments at different elevations may beanalyzed from images. The use of multiple segments at differentelevations can also be used to generate a three-dimensionalrepresentation of temperature information.

Embodiments within the scope of the present application include programproducts comprising machine-readable media for carrying or havingmachine-executable instructions or data structures stored thereon. Suchmachine-readable media can be any available media that can be accessedby a general purpose or special purpose computer or other machine with aprocessor. By way of example, such machine-readable media can compriseRAM, ROM, EPROM, EEPROM, CD-ROM or other optical disk storage, magneticdisk storage or other magnetic storage devices, or any other mediumwhich can be used to carry or store desired program code in the form ofmachine-executable instructions or data structures and which can beaccessed by a general purpose or special purpose computer or othermachine with a processor. When information is transferred or providedover a network or another communications connection (either hardwired,wireless, or a combination of hardwired or wireless) to a machine, themachine properly views the connection as a machine-readable medium.Thus, any such connection is properly termed a machine-readable medium.Combinations of the above are also included within the scope ofmachine-readable media. Machine-executable instructions comprise, forexample, instructions and data which cause a general purpose computer,special purpose computer, or special purpose processing machines toperform a certain function or group of functions.

The method of operating the furnace further comprises providing anestimate of a mathematical function characterizing a relationshipbetween burner flow rate changes for two or more of the plurality ofburners and individual temperature changes for at least a portion of theplurality of process tubes. Each burner may have several flows that passthrough it. These include both fuel streams and oxidant streams and itmay be possible to control the flows of these streams to individualburners or to proper subsets of burners (such as a single row ofburners). Altering the flow of one of these streams (by for example,increasing or decreasing the fractional opening of a valve in the flowpath) will affect the temperatures of one or more tubes in the furnace.Such a change is not intended to affect the overall flows of fuels andoxidants to the furnace as a whole, only the distribution of such flowswithin the furnace. In general terms, the relationship between a singleflow rate change (Δu_(i) for stream i, where i is an index specifyingeach stream in the plurality of streams which affect the distribution offuel or oxidants within the furnace; the total number of such streams isdesignated N_(stream)) and the change in temperatures of the tubes(ΔT_(j) for all j where j is an index specifying each of tube in theplurality of tubes in the furnace which are considered in thecalculation) is represented as

Δ T=f (Δu _(i)),

where the underbar is used to denote a vector quantity. In this case, avector of length N_(tube), the number of tubes of the portion of theplurality of tubes in the furnace that are controlled (and which may,but does not have to, equal the total number of tubes in the furnace).So this vector equation represents the N_(tube) individual generalfunctional relationships between the flow rate change of a single fuelor oxidant stream and the temperature of the N_(tube) tubes.

It is often convenient to balance the furnace in stages where a singlefuel or oxidant stream is adjusted in each stage. The streams may thenbe designated a specific order in which they are to be used forbalancing. If a burner has 3 fuel streams, they could be designatedfuel1,fuel2 and fuel3. If there are 2 oxidant streams they could bedesignated oxidant1 and oxidant2. The total number of stream types maybe designated M_(streams). During the first stage, fuel1, by itself, maybe adjusted to balance the furnace. All the balancing may be done in asingle stage (i.e., using just one stream for balancing even if multipleadjustment steps or iteration are required) or subsequent stages may usethe other streams to further improve the furnace balance. In eithercase, when using the staged approach the only functional relationshipsneeded within a stage are those between the change in tube temperatures(ΔT_(j)) and the change of a single stream type as it flows through theplurality of burners (Δu_(I)). The upper case index I is used todesignate the burner through which the specific stream flows and runsfrom 1 to the number of the plurality of burners (N_(burner))considered. In general the total number of streams flowing through allthe burners is equal to the product of the number of stream types andthe number of burners (i.e., N_(stream)=M_(stream)×N_(burner)). It isalso possible to define a one-to-one correspondence between the index iused to designate any stream and a compound index I_(m) where Idesignates the burner and the subscript m designates the stream type. Inthis way stream i is associated with burner I and if the method isapplied in the stage-wise manner, the indexes are identical and used toidentify a specific burner stream. The lower case index will be use todesignate a burner stream and the upper case index the correspondingburner through which it flows. Equations for the stage-wise approachcould be generated in a more explicit form by replacing the lower case iwith the upper case I.

Any function can be linearized so that for small changes in theindependent variable a constant value accurately represents the changein the dependent variable. This is the first derivative and in this casemay be represented as

$\frac{\partial T_{j}}{\partial u_{i}}.$

Here the symbol ∂ is used to represent the partial derivative since thetemperature of the tube wall is taken as a function of many independentvariables (the u_(i)'s, which are the controllable stream flows). Thesymbol g_(ij) is defined herein as the estimate of

$\frac{\partial T_{j}}{\partial u_{i}}.$

The function which represents the relationship between two or more ofthe plurality of flow streams to the burner and the tube walltemperatures is represented as

Δ T=f (Δ u )

This is shorthand for a series of N_(tube) equations each of whichprovides the change in tube wall temperature for a specific tube as afunction of changes to all the flow streams. A single exemplary equationmay be written as

ΔT _(j) =f _(i)(Δ u )=f _(j)(Δu ₁ ,Δu ₂ ,Δu ₃ , . . . ,Δu _(N)_(streams) ).

After linearizing the function, the single exemplary equation may bewritten as

${\Delta \; T_{j}} = {\sum\limits_{\forall i}{\frac{\partial T_{j}}{\partial u_{i}}\Delta \; {u_{i}.}}}$

After replacing

$\frac{\partial T_{j}}{\partial u_{i}}$

with the estimated value g_(i,j), the single exemplary function may bewritten as

ΔT_(j)=Σ_(∀i)g_(ij)Δu_(i)

There are N_(tube) of these individual equations, one for each of theportion of the plurality of tubes in the furnace which are controlled.Together these N_(tube) equations maybe written in shorthand as

ΔT=GΔu

where G is a matrix with functional elements g_(i,j) and dimensionsN_(stream)×N_(tube).

It remains to determine values for g_(i,j). This can be arduous,repetitive work if tackled experimentally. However, there are featuresin a typical furnace that can be exploited to simplify the task.Specifically, tubes that are near a burner in which the fuel flow isincreased see an increase in the temperature and a larger increase thanthose that are more distant. Tubes that are remote from the burner thatsees an increase in fuel may see no change or even a decrease intemperature as burners nearer the remote tube experience a slightdecrease in fuel flow (as the total fuel flow to the furnace isunchanged). This suggests certain functional forms for the functionalelements g_(i,j) which provide estimates of

$\frac{\partial T_{j}}{\partial u_{i}}.$

For example, the functional form

${g_{i,j} = {{a_{1} \times {\exp \left( \frac{a_{2}}{d_{I,j}} \right)}} + a_{3}}},$

where d_(I,j) is the distance between tube j and burner I (note that theupper case I is used to designate the burner through which stream iflows as described previously) may be used. This form has threeparameters (a₁, a₂ and a₃). In the most general sense, parameters couldbe determined for each tube/stream pair and as functions of operatingand environmental conditions, but this approach does not exploit thepower of this methodology. Another approach is to assume that theparameters (a₁, a₂ and a₃) are valid for every tube/stream pair withinspecific classes. For example burners may have two separate fuel streamsand one set of parameters (a₁, a₂ and a₃) may be determined for eachfuel stream separately. Burners within the furnace may be of differentsizes and a set of parameters may be determined for each fuel stream foreach size burner. Since the estimates need not be perfect, it may bedesirable to limit the number of different classes and accept a level ofinaccuracy in the estimated response rather than devote additional workto parameter determination for more classes.

Therefore, in the method, the estimate of the mathematical function(e.g. ΔT=GΔu) may comprise calculated values from functional elements(g_(i,j)) wherein each of the functional elements (g_(i,j)) comprise afunctional form (e.g.

$\left. {g_{i,j} = {{a_{1} \times {\exp \left( \frac{a_{2}}{d_{I,j}} \right)}} + a_{3}}} \right)$

comprising a first functional parameter, a₁, a second functionalparameter, a₂, and a geometric parameter, d_(I,j), characterizingdistances between each of the plurality of process tubes and each of theplurality of burners; wherein the first functional parameter, a₁, of afirst functional element of the functional elements has the same valueas the first functional parameter a₁, of a second functional element ofthe functional elements and wherein the second functional parameter, a₂,of a first functional element of the functional elements has the samevalue as the second functional parameter, a₂, of a second functionalelement of the functional elements.

If g_(i,j) is non-dimensionalized by dividing by a reference stream flowchange and multiplying by a reference tube temperature change, andd_(I,j) is non-dimensionalized by dividing by a furnace length scalethen a set of parameter values that has the form described above isa₁=−1.1, a₂=−0.05 and a₃=1 (these values have been chosen as arepresentative example and are not intended to describe any specificfurnace nor to limit the method in any way). FIG. 4 is a plot of thisexample function with the given parameter values. With dimensionlessvalues, a₃ may be appropriately chosen to equal 1. Larger absolutevalues of a₂ restrict the direct effect of flow changes to a smallerneighborhood around the burner. a₁ may be chosen to be less than −1.More complex forms of the functional elements may also be selected, forexample,

$g_{i,j} = {{a_{1}{\tan^{- 1}\left( {\frac{a_{2}}{d_{I,j}} + \frac{a_{3}}{d_{I,j}^{2}}} \right)}} + {a_{4}.}}$

Here, the functional element g_(i,j) has 4 parameters and this functioncan more closely match experimental data. Other functional forms may bechosen as well.

The parameter values (a_(k)) used to determine the value of thefunctional elements for each specific class can be updated as additionalinformation regarding the effect of burner stream flow rate changes ontube temperature changes is obtained. This is done by performing thefollowing minimization:

$\underset{\forall a_{k}}{Min}{{\left( {{\Delta \; \underset{\_}{T}} - {\underset{\underset{\_}{\_}}{G}\; \Delta \; \underset{\_}{u}}} \right) \cdot \underset{\_}{ɛ^{- 1}}}}$

where ε⁻¹ is a vector (of dimension N_(tube)) in which each element isthe reciprocal of the uncertainty in the tube temperature measurement,the ° operator is used for the point-wise product (a.k.a. Hadamardproduct) of the two vectors. If matrix G is ill conditioned, it may bereconditioned by dropping some of the smaller eigenvalues.

As indicated above, the relationship between tube wall temperaturechanges and changes in the burner flow may be affected by operatingconditions (e.g., production rate) or environmental conditions (e.g.,ambient temperature). These effects can be captured by changing theparameter values (a_(k)). However, the values determined for previousconditions provide a reasonable estimate and a good starting point forthe mathematical function relating tube wall temperature changes tochanges in the burner flows. As changes are made to the burner streamflows, the parameters may be updated as described above.

The method of operating a furnace further comprises calculating a firsttarget flow rate solution set having solutions for the two or more ofthe plurality of burners consistent with conforming temperatures of theplurality of process tubes to the temperature uniformity limitation forthe plurality of process tubes. Mathematically, this is done by firstconstructing a vector representing the difference between the currenttube temperatures (labeled T_(j)∀j or simply T) and the targettemperatures (T_(j)* for ∀j or simply T*) of the form T−T* (or ΔT*) andthen determining the solution (Δu) which conforms to the linearizedfunction which estimates the relationship between changes in tube walltemperatures and burner flows subject to the constraint that thedetermined values (Δu_(i)) lie between the lower and upper bounds on theindividual burner flows. Symbolically this is

$\underset{\underset{{s.t.{LB}_{i}} \leq {\Delta \; u_{i}} \leq {{UB}_{i}{\forall i}}}{\Delta \; \underset{\_}{u}}}{Min}{{{\Delta \; {\underset{\_}{T}}^{*}} - {\underset{\underset{\_}{\_}}{G}\; \Delta \; \underset{\_}{u}}}}$

where the generic symbol λxλ is used to designate any of the variety ofoptimization targets such as the 2-norm discussed previously. T_(j)* mayequal T or be related to T as described previously. G may bereconditioned.

The solution (Δu) is a set of burner flow changes that minimize thedifference between the estimated tube wall temperatures and the targettemperatures. These conforming flow changes can be implemented as valveadjustments. When this is done a new temperature profile will appear inthe furnace. The tube temperatures can be measured as described abovealong with corresponding uncertainty values. The new temperatureinformation may then be used to estimate new values for the parametersin g_(i,j), as well as determine a new ΔT* so that the process may berepeated to further refine the temperature profile.

The target flow rate solution set may be any measure relating to burnerflow rates, e.g. specific oxidant gas and/or fuel flow rates, changes inflow rates, percent opening/closing of valves, etc. Conforming may be byany suitable technique for decreasing the difference between themeasured temperatures and the temperature target.

As described above, the temperature values or temperature informationmay include uncertainty values (ε_(j)). To incorporate the uncertaintyin tube wall temperature, the minimization process by which theconforming flows are determined is modified to include a weighting ofthe individual vector elements with the reciprocal of the uncertainty.This is similar to how the uncertainty was used to compute the parametervalues in the estimate of the individual function elements.Symbolically, this is

$\underset{\underset{{s.t.{LB}_{i}} \leq {\Delta \; u_{i}} \leq {{UB}_{i}{\forall i}}}{\Delta \; \underset{\_}{u}}}{Min}{{{{\Delta \; {\underset{\_}{T}}^{*}} - {\underset{\underset{\_}{\_}}{G}\; \Delta \; {\underset{\_}{u} \cdot ɛ^{- 1}}}}}.}$

The modification encompasses differentiation in uncertainty that may ormay not be present in different tube wall temperatures. The resultingsolution puts more emphasis on moving valves when the uncertaintyassociated with the related temperature is low compared to cases whenthe uncertainty is higher.

The most obvious constraints on burner flows are minimum and maximumflows. These could be specified to maintain some minimum fuel flowconsistent with burner stability or maximum fuel flow associated withfuel-rich combustion and emissions limits. These flow constraints may berecast in terms of valve position constraints to be consistent with thecase in which the Δu_(i)'s are considered to be valve position changes.There are also additional constraints which may be included. For examplea constraint could be imposed on the number of valves that are allowedto be manipulated for each step of calculating a target flow ratesolution set.

The method of operating the furnace 102 further comprises adjusting afirst valve upstream of at least one of the two or more of the pluralityof burners 108 to change at least one of the flow rates of the two ormore of the plurality of burners 108 in accordance with the first targetflow rate solution set. Reference to a first valve includes one or morevalves since the article “a” means one or more when applied to anyfeature. The first valve may be a valve controlling the flow of oxidantgas or a valve controlling the flow of fuel gas. The first valve may bean actuated valve or manual valve. The first valve that is adjusted isnot a main oxidant gas or main fuel valve for controlling the total flowto the entire furnace which is controlled by traditional methods ratherit is a valve that controls the flow to a proper subset of the burnersand therefore affects the distribution of flow.

The method of operating the furnace 102 may further comprise measuringsecond temperature information for the plurality of process tubes at asecond operating condition different from the first operating condition.The second operating condition is subsequent (at a later time) to thefirst operating condition.

The second temperature information may be measured in a manner similarto the first temperature information.

In combination with any of the other features, the second temperatureinformation of the plurality of process tubes may be measured bycapturing a second plurality of images of an interior area of thereformer furnace, at least some images of the second plurality of imagesbeing associated with different portions of the interior area of thereformer furnace, wherein each image of the second plurality of imagescomprises second pixel data associated with a respective portion of theplurality of process tubes, and processing a portion of the second pixeldata to obtain the second temperature information for the plurality ofprocess tubes.

The second operating condition may be the result of open loop testing ofthe temperature response of oxidant gas/fuel flow rates to the burners.Open loop testing includes the case where a single burner flow stream isadjusted specifically for the purpose of determining the parameters usedto define the functional elements for a specific class of burner flows.The parameter estimates may be calculated using the first temperatureinformation and the second temperature information resulting from theburner flow rate change(s). If ₀ T is used to designate the measuredtemperatures at the initial condition and ₁ T is used to designate themeasured temperatures at the subsequent condition then ΔT=₁ T−₀ T andthe parameters are determined from

$\underset{\forall a_{k}}{Min}{{\left( {{\Delta \; \underset{\_}{T}} - {\underset{\underset{\_}{\_}}{G}\; \Delta \; \underset{\_}{u}}} \right) \cdot \underset{\_}{ɛ^{- 1}}}}$

where the elements of ε⁻¹ are defined as

$ɛ_{i}^{- 1} = \frac{1}{\sqrt{{{}_{}^{}{}_{}^{}} + {{}_{}^{}{}_{}^{}}}}$

and Δu is a vector with only one non-zero element (Δu_(j)) correspondingto the burner flow stream that was adjusted for the open loop test. Anynumber of open loop tests may be conducted to obtain more temperatureinformation (₂ T, ₃ T, . . . ) to validate the form of the functionalelement chosen for g_(i,j), refine the parameter estimates, to developparameter estimates for additional classes of burner flows or atdifferent operating conditions.

Alternatively to open loop testing, the second operating condition wherethe second temperature information is measured may be the result ofadjusting the first valve in accordance with the first target flow ratesolution set. With each adjustment of one or more valves, additionaltemperature information may be measured and the results used to updatethe parameter estimates.

The method of operating the furnace 102 may further comprise calculatinga second target flow rate solution set having solutions for the two ormore of the plurality of burners consistent with conforming temperaturesof the plurality of process tubes to the target temperature criterionusing the estimate or an updated estimate of the mathematical functionand using the second temperature information for the plurality ofprocess tubes. The estimate of the mathematical function or an updatedestimate of the mathematical function is evaluated using the valuesprovided in the second temperature information to calculate the secondtarget flow rate solution set. The second target flow rate solution sethas updated or second solutions for the two or more of the plurality ofburners. The same estimate of the mathematical function used previouslycould be used to calculate the second target flow rate solution set oran updated estimate of the mathematical function could be provided basedon the new temperature information. In case an updated estimate of themathematical function is used, the estimate of the mathematical functionis updated from the second temperature information. The mathematicalfunction is updated by re-estimating the parameters of the functionalelement (a_(k)) using the new information from the second temperaturedata. In the case that there are multiple conditions resulting inmultiple temperature readings and multiple valve positions, the resultis combined by

${\underset{\forall a_{k}}{Min}{\sum\limits_{\forall l}{{\left( {{\Delta_{l}\underset{\_}{T}} - {\underset{\underset{\_}{\_}}{G}\; \Delta_{l}\underset{\_}{u}}} \right) \cdot \underset{\_}{{}_{}^{}{}_{}^{- 1}}}}}},$

where l is the index indentifying each of the different conditionsevaluated.

Constraints related to minimum and/or maximum allowable valve positionsmay be taken into account for estimating the next control action. Theconstraint may be a physical constraint (i.e. full open or full closedvalve). The constraint may be based on experience that a valve shouldnot be open or closed beyond a certain position. Other constraints maybedue to total number of burners allowed to be moved at each iteration, orthe total number of valves allowed to be closed, or maximum change inback pressure that is allowed.

After the second target flow rate solution set is calculated, the methodof operating the furnace may then further comprise adjusting the firstvalve or a second valve upstream of at least one of the two or more ofthe plurality of burners to change at least one of the flow rates of thetwo or more of the plurality of burners in accordance with the secondtarget flow rate solution set. The first valve or the second valve thatis adjusted is not a main oxidant gas or main fuel valve for controllingthe total flow to the entire furnace which is controlled by traditionalmethods rather it is a valve that controls the flow to a proper subsetof the burners and therefore affects the distribution of flow.

The furnace may be operated to first adjust the header valves regulatingeach row of burners followed by adjusting individual burner valvesregulating each individual burner.

The plurality of burners may comprise two or more rows of burners andthe first valve that is adjusted in response to the first target flowrate solution set may be upstream of a first row of burners. A row ofburners is a plurality of burners connected to a common header andhaving outlets arranged in a substantially straight line. The secondvalve that is adjusted in response to the second target flow ratesolution set may be upstream of a single burner of the two or more ofthe plurality of burners to change at least one of the flow rates of thesingle burner in accordance with the second target flow rate solutionset. Since the second valve regulates only a single burner, the secondvalve is not upstream of any burner other than the single burner.

Defining classes of burner flow streams and representing the change intube wall temperature to changes in burner flow streams with a uniqueexpression for each class is a powerful tool which exploits the regulargeometric pattern of the furnace and allows the gain matrix to be morefully populated with relatively few perturbations (i.e., one for eachclass). FIG. 1 shows that tubes 14 which surround burner 16 are in asimilar relationship as tubes 24 which surround burner 26. It isexpected that tubes 24 will respond to changes in burner 26 in much thesame manner that tubes 14 respond to changes in burner 14. Thisexpectation has been verified experimentally. Likewise, defining eachelement of the gain matrix based on a functional form that is related togeometric considerations (the distance between the burner and the tube)further enhances the efficiency of the method. This ensures thatrelatively minor effects on distant tubes are considered even if onlyapproximately. The sum of the minor effects can be significant, soignoring these altogether makes the ultimate solution of this largedimensional problem less efficient.

In an embodiment, after calculating a target flow rate solution set,valves upstream of each burner may be adjusted one at a time,temperature information measured after each adjustment, and thetemperature information measured after each adjustment used to updatethe estimate to the mathematical function. The update to the estimate ofthe mathematical function may be made after some or all of the earlierprescribed changes of the previous target flow rate solution set havebeen made. This approach has the benefit of more readily improving theestimate of the mathematical function.

In this embodiment, the first valve is upstream of a lone first burnerof the two or more of the plurality of burners. The method may thenfurther comprise measuring second temperature information for theplurality of process tubes at a second operating condition wherein thesecond operating condition results from adjusting the first valve,adjusting a second valve in accordance with the first target flow ratesolution set wherein the second valve is upstream of a lone secondburner of the two or more of the plurality of burners, measuring thirdtemperature information for the plurality of process tubes at a thirdoperating condition wherein the third operating condition results fromadjusting the second valve in accordance with the first target flow ratesolution set, updating the estimate of the mathematical function fromthe second temperature information and the third temperature informationthereby forming the updated estimate of the mathematical function,calculating a second target flow rate solution set having solutions forthe two or more of the plurality of burners consistent with conformingtemperatures of the plurality of process tubes to the target temperaturecriterion using the updated estimate of the mathematical function; andadjusting at least one of the first valve, the second valve or a thirdvalve upstream of the two or more of the plurality of burners to changeat least one of the flow rates of the two or more of the plurality ofburner in accordance with the second target flow rate solution setwherein the third valve is not upstream of all burners in the reformerfurnace.

The present invention will be better understood with reference to thefollowing example, which is intended to illustrate, but not to limit thescope of the invention. The invention is solely defined by the claims.

Example 1

This example illustrates the method in practice.

Step 1. The target temperature criterion for uniform tube temperatureswithin the furnace, w∥(T− T)∥₂+(1−w)∥(T− T)∥_(∞) was chosen. Thiscriterion is the weighted sum of the 2-norm and ∞-norm differencebetween the individual tube temperature and the average of all tubetemperatures which are recorded.

Step 2. Temperature information at an initial condition, T_(j),comprising data for each of the plurality of process tubes j wasmeasured using a modified digital camera, where j is 1 through the totalnumber of process tubes visible in the furnace, N_(tube). In this caseover 90% of the tubes were visible in the images. The temperatureinformation for the plurality of process tubes was measured by capturinga plurality of images (taking a “picture”) of an interior area of thereformer furnace, at least some images of the plurality of images beingassociated with different portions of the interior area of the reformerfurnace, wherein each image of the plurality of images comprises pixeldata associated with a portion of the plurality of process tubes. Thetemperature information for the plurality of process tubes was thenobtained by processing a portion of the pixel data. “Pictures” of thetubes were taken and a correlation used to convert the pictures to atemperature value and respective uncertainty. The estimate of theuncertainty in the tube wall temperature was provided by the standarddeviation of the estimate of the tube wall temperature and designated asε_(j). The temperatures T_(j), together at the initial condition form atemperature vector which is designated ₀ T. The uncertainties ε_(j)together at the initial condition form an uncertainty vector ₀ ε. Theinitial temperature data showed a temperature spread of over 50° C.,which is consistent with the temperature spread achievable using priorart techniques.

Step 3. An estimate of a mathematical function, ΔT=GΔu, was made usingprior knowledge. The mathematical function characterizes a relationshipbetween the changes in tube wall temperature and the changes in thevalve position controlling the flow of fuel1 to an individual burner. Gis a matrix with functional elements g_(I,j) and dimensionsN_(burner)×N_(tube).

The form of the functional elements g_(j1) in this example was

g _(I,j) =a ₁×exp(a ₂ d _(I,j))+a ₃

where the functional elements g_(I,j) are the gains for a change invalve position (measured in ° C./percent valve opening) associated withfuel stream fuel1 in burner I and reactor tube j; d_(I,j) is thedistance from reactor tube j to burner I (measured in meters) and a₁,a₂, and a₃, are parameters which correlate the relationship between theburner flow rates and the reactor tube temperatures. The initial valuesof a₁, a₂, and a₃ were 45(° C./%), −2.3 m⁻¹, 0.5(° C./%), respectively.The initial values of a₁, a₂, and a₃ were provided based on estimates.The valve position is related to the fuel stream flow via a valve curve(which describes the resistance of the valve), the pressure differentialand the physical properties of the fluid. Mathematically this conversionis achieved by application of the chain rule. In this example, theactual stream flow change was not computed; rather the change in valveposition which affects the flow was the determined variable.

Step 4. In this furnace, there are two different sized burners. The siderow burners (those adjacent to a refractory wall) are only 65% as largeas the burners in the interior of the furnace. A percentage change inthe valve position of a side row burner has only ˜65% as much change inthe fuel stream flow (and therefore in tube wall temperature) as asimilar change in an interior row burner. To account for this all thegain elements associated with side row burners are multiplied by thisduty ratio, 0.65. Then the modified gains, g_(I,j)′, for the sideburners are:

g_(I,j)′=0.65g_(I,j),

for every I corresponding to a side burner.

Step 5. The measured tube temperatures, T, uncertainty vector ε, thecurrent estimate of G, and a weighting function w were used to estimatethe changes in valve position that will determine the flow of fuel1 toeach burner, Δu, that best satisfied the target temperature criterion.The chosen target temperature criterion contains a weighting factor, wwhich was initially set to 1.0. We have found that by starting with w=1and finishing with w=0.5 the furnace balance converges more quickly thanwith a constant value of w. The change in the valve position whichcontrols the flow of fuel stream fuel1 through each burner is calculatedsuch that

${\underset{\underset{{s.t.{LB}_{i}} \leq {\Delta \; u_{i}} \leq {{UB}_{i}{\forall i}}}{\Delta \; \underset{\_}{u}}}{Min}\left( {w{{\left( {T - \overset{\_}{T} - {\underset{\underset{\_}{\_}}{G}\; \Delta \; \underset{\_}{u}}} \right) \cdot \underset{\_}{ɛ^{- 1}}}}_{2}} \right)} + \left( {\left( {1 - w} \right){{\left( {\underset{\_}{T} - \overset{\_}{T} - {\underset{\underset{\_}{\_}}{G}\; \Delta \; \underset{\_}{u}}} \right) \cdot \underset{\_}{ɛ^{- 1}}}}_{\infty}} \right)$

subject to a limitation for a lower bound, LB, of 20% open and an upperbound, UB, of 100% open for each Δu_(i). We have found that including anadditional constraint that limits the number of valve changes to 5allows for easier practical implementation. improves the stability ofthe convergence and eliminates inconsequential small valve changes. Thisset of valve changes which satisfies the minimization is the conformingsolution.

Step 6. The 5 valve changes associated with the conforming solution aremade to the burner valves. After the effect of the burner stream flowrate changes on tube temperatures occurred (we waited a minimum of 2hours at each iteration), additional tube temperature informationcomprising data for each of the tubes was collected using the modifieddigital camera like in step 2. “Pictures” of the tubes were taken and acorrelation used to convert the pictures to temperature values at asecond condition, represented as a vector, ₁ T, with correspondinguncertainty vector ₁ ε for each of the plurality of process tubes.

Step 7. After the initial valve move the temperature variation wasreduced to less than 50° C., but the hottest tube was still above thetemperature needed for maximum tube life. So, the method was continuedby first updating the parameters a₁, a₂, and a₃ according to step 8.

Step 8. Given the new temperature data, ₁ T, with correspondinguncertainty ₁ ε, the previous temperature data ₀ T, with correspondinguncertainty ₀ ε, and corresponding burner valve changes, Δu, parametersa₁, a₂, and a₃ were re-estimated. Parameters a₁, a₂, and a₃ arere-estimated according to

$\underset{a_{1},a_{2},a_{3}}{Min}{{{\left( {\left( {{\,_{1}\underset{\_}{T}} - {\,_{0}\underset{\_}{T}}} \right) - {\underset{\underset{\_}{\_}}{G} \times \Delta \; \underset{\_}{u}}} \right) \cdot \left( \sqrt{\left( {{{}_{}^{}{ɛ\_}_{}^{}} + {{}_{}^{}{ɛ\_}_{}^{}}} \right)} \right)^{- 1}}}.}$

For the first update, only one data set is available for re-estimatingthe parameters a₁, a₂, and a₃ as shown in this equation. At subsequentsteps, all previous time steps were included as discussed previously.

Steps 3 through 8 are repeated using the updated parameters a₁, a₂, anda₃ at each iteration. After 3 iterations the maximum tube temperaturewas less than 15° C. above the average and the total variation is ˜40°C. The final values of parameters a₁, a₂, and a₃ were 44.4(° C./%),−0.23(° C./%), respectively.

While only certain features and embodiments of the invention have beenshown and described, many modifications and changes may occur to thoseskilled in the art (for example, variations in sizes, dimensions,structures, shapes and proportions of the various elements, values ofparameters (for example, temperatures, pressures, etc.), mountingarrangements, use of materials, colors, orientations, etc.) withoutmaterially departing from the novel teachings and advantages of thesubject matter recited in the claims. The order or sequence of anyprocess or method steps may be varied or re-sequenced according toalternative embodiments. It is, therefore, to be understood that theappended claims are intended to cover all such modifications and changesas fall within the true spirit of the invention. Furthermore, in aneffort to provide a concise description of the exemplary embodiments,all features of an actual implementation may not have been described(i.e., those unrelated to the presently contemplated best mode ofcarrying out the invention, or those unrelated to enabling the claimedinvention). It should be appreciated that in the development of any suchactual implementation, as in any engineering or design project, numerousimplementation specific decisions may be made. Such a development effortmight be complex and time consuming, but would nevertheless be a routineundertaking of design, fabrication, and manufacture for those ofordinary skill having the benefit of this disclosure, without undueexperimentation.

1. A method of operating a furnace having a plurality of burners, eachof the plurality of burners having flow rates associated therewith, thefurnace containing a plurality of process tubes, the method comprising:(a) selecting target temperature criterion for the plurality of processtubes; (b) measuring first temperature information comprising data foreach of the plurality of process tubes at a first operating conditionwherein the first temperature information for the plurality of processtubes is measured by: capturing a first plurality of images of aninterior area of the reformer furnace, at least some images of the firstplurality of images being associated with different portions of theinterior area of the reformer furnace, wherein each image of the firstplurality of images comprises first pixel data associated with a portionof the plurality of process tubes; and processing a portion of the firstpixel data to obtain the first temperature information for the pluralityof process tubes; (c) providing an estimate of a mathematical functioncharacterizing a relationship between burner flow rate changes for twoor more of the plurality of burners and individual temperature changesfor at least a portion of the plurality of process tubes; (d)calculating a first target flow rate solution set having solutions forthe two or more of the plurality of burners consistent with conformingtemperatures of the plurality of process tubes to the target temperaturecriterion for the plurality of process tubes using the estimate of themathematical function and the first temperature information for theplurality of process tubes; and (e) adjusting a first valve upstream ofat least one of the two or more of the plurality of burners to change atleast one of the flow rates of the two or more of the plurality ofburners in accordance with the first target flow rate solution setwherein the first valve is not upstream of all burners of the furnace.2. The method of claim 1 further comprising: measuring secondtemperature information for the plurality of process tubes at a secondoperating condition different from the first operating condition; andwherein the estimate of the mathematical function provided in step (c)is calculated using the first temperature information and the secondtemperature information.
 3. The method of claim 2 wherein the secondtemperature information for the plurality of process tubes is measuredby: capturing a second plurality of images of the interior area of thereformer furnace, at least some images of the second plurality of imagesbeing associated with different portions of the interior area of thereformer furnace, wherein each image of the second plurality of imagescomprises second pixel data associated with a portion of the pluralityof process tubes; and processing a portion of the second pixel data toobtain the second temperature information for the plurality of processtubes.
 4. The method of claim 3 wherein the first temperatureinformation includes uncertainty values and the second temperatureinformation includes uncertainty values; and wherein the estimate of themathematical function and/or the updated estimate of the mathematicalfunction are calculated using the uncertainty values of the firsttemperature information and the uncertainty values of the secondtemperature information.
 5. The method of claim 1 further comprising:measuring second temperature information for the plurality of processtubes at a second operating condition different from the first operatingcondition and subsequent to the first operating condition; calculating asecond target flow rate solution set having solutions for the two ormore of the plurality of burners consistent with conforming temperaturesof the plurality of process tubes to the target temperature limitationsusing the estimate or an updated estimate of the mathematical functionand using the second temperature information for the plurality ofprocess tubes; and adjusting the first valve or a second valve upstreamof at least one of the two or more of the plurality of burners to changeat least one of the flow rates of the two or more of the plurality ofburners in accordance with the second target flow rate solution setwherein the second valve is not upstream of all burners of the reformerfurnace.
 6. The method of claim 5 wherein the second operating conditionresults from step (e).
 7. The method of claim 5 further comprising:updating the estimate of the mathematical function from the secondtemperature information for the plurality of process tubes therebyforming the updated estimate of the mathematical function; and whereinthe step of calculating the second target flow rate solution set usesthe updated estimate of the mathematical function.
 8. The method ofclaim 1 wherein the estimate of the mathematical function comprisescalculated values from functional elements wherein each of thefunctional elements comprise a functional form comprising a firstfunctional parameter, a second functional parameter and a geometricparameter characterizing distances between each of the plurality ofprocess tubes and each of the plurality of burners; wherein the firstfunctional parameter of a first functional element of the functionalelements has the same value as the first functional parameter of asecond functional element of the functional elements; and wherein thesecond functional parameter of a first functional element of thefunctional elements has the same value as the second functionalparameter of a second functional element of the functional elements. 9.The method of claim 8 wherein the first functional parameter has a valuethat is the same for each of the functional elements and wherein thesecond functional parameter has a value that is the same for each of thefunctional elements.
 10. The method of claim 1 wherein the plurality ofburners comprises two or more rows of burners and wherein the firstvalve is upstream of a first row of burners, the method furthercomprising: measuring second temperature information for the pluralityof process tubes at a second operating condition different from thefirst operating condition and subsequent to the first operatingcondition; calculating a second target flow rate solution set havingsolutions for the two or more of the plurality of burners consistentwith conforming temperatures of the plurality of process tubes to thetarget temperature limitations using the estimate or an updated estimateof the mathematical function and using the second temperatureinformation for the plurality of process tubes; and adjusting a secondvalve upstream of a single burner of the two or more of the plurality ofburners to change at least one of the flow rates of the single burner inaccordance with the second target flow rate solution set wherein thesecond valve is not upstream of any burner other than the single burner.11. The method of claim 1 wherein the first valve is upstream of a lonefirst burner of the two or more of the plurality of burners, the methodfurther comprising: measuring second temperature information for theplurality of process tubes at a second operating condition wherein thesecond operating condition results from step (e); adjusting a secondvalve in accordance with the first target flow rate solution set whereinthe second valve is upstream of a lone second burner of the two or moreof the plurality of burners; measuring third temperature information forthe plurality of process tubes at a third operating condition whereinthe third operating condition results from adjusting the second valve inaccordance with the first target flow rate solution set; updating theestimate of the mathematical function from the second temperatureinformation and the third temperature information thereby forming theupdated estimate of the mathematical function; calculating a secondtarget flow rate solution set having solutions for the two or more ofthe plurality of burners consistent with conforming temperatures of theplurality of process tubes to the target temperature limitations usingthe updated estimate of the mathematical function; and adjusting atleast one of the first valve, the second valve or a third valve upstreamof the two or more of the plurality of burners to change at least one ofthe flow rates of the of the two or more of the plurality of burner inaccordance with the second target flow rate solution set wherein thethird valve is not upstream of all burners in the reformer furnace. 12.The method of claim 1 wherein the estimate of the mathematical functionis represented asΔT=GΔu where ΔT represents the individual temperature changes for the atleast a portion of the plurality of process tubes, Δu represents burnerflow rate changes for the two or more of the plurality of burners and Gis a gain matrix.